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M110 Fa17
Worksheet 18 - Inverse Trigonometric Functions (§7.4)
In Exercises 1 - 40, compute the exact value.
√ !
3
1. arcsin (−1)
2. arcsin −
2
1
6. arcsin
2
5. arcsin (0)
9. arcsin (1)
14. arccos (0)
√ !
3
2
18. arccos (1)
1
13. arccos −
2
17. arccos
10. arccos (−1)
√ !
3
21. arctan −
3
22. arctan (0)
"√ 25. arctan 3
" √ 26. arccot − 3
29. arccot (0)
30. arccot
33. arcsec (2)
34. arccsc (2)
37. arcsec
√ !
2 3
3
38. arccsc
√ !
3
3
√ !
2 3
3
√ !
2
3. arcsin −
2
√ !
2
7. arcsin
2
√ !
3
11. arccos −
2
1
15. arccos
2
" √ 19. arctan − 3
23. arctan
√ !
3
3
1
4. arcsin −
2
√ !
3
8. arcsin
2
√ !
2
12. arccos −
2
√ !
2
16. arccos
2
20. arctan (−1)
24. arctan (1)
27. arccot (−1)
√ !
3
28. arccot −
3
31. arccot (1)
32. arccot
"√ 3
36. arccsc
"√ 2
35. arcsec
"√ 2
39. arcsec (1)
40. arccsc (1)
In Exercises 41" - 48,
assume
that the range of arcsecant is 0, π2 ∪ π, 3π
and that the range of
2
"
π
3π
arccosecant is 0, 2 ∪ π, 2 when computing the exact value.
√ !
" √ 2 3
41. arcsec (−2)
42. arcsec − 2
44. arcsec (−1)
43. arcsec −
3
√ !
" √ 2 3
48. arccsc (−1)
45. arccsc (−2)
46. arccsc − 2
47. arccsc −
3
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M110 Fa17
π "π In Exercises 49 - 56, assume
that
the
range
of
arcsecant
is
0, 2 ∪ 2 , π and that the range of
"
arccosecant is − π2 , 0 ∪ 0, π2 when computing the exact value.
√ !
" √ 2 3
49. arcsec (−2)
50. arcsec − 2
52. arcsec (−1)
51. arcsec −
3
√ !
" √ 2 3
53. arccsc (−2)
54. arccsc − 2
55. arccsc −
56. arccsc (−1)
3
In Exercises 57 - 86, compute the exact value or state that it is undefined.
√ !!
1
2
3
57. sin arcsin
58. sin arcsin −
59. sin arcsin
2
2
5
√ !!
2
5
60. sin (arcsin (−0.42))
61. sin arcsin
62. cos arccos
4
2
1
5
63. cos arccos −
64. cos arccos
65. cos (arccos (−0.998))
2
13
"
"√ 66. cos (arccos (π))
67. tan (arctan (−1))
68. tan arctan 3
5
70. tan (arctan (0.965))
71. tan (arctan (3π))
69. tan arctan
12
"
" √ 7
72. cot (arccot (1))
73. cot arccot − 3
74. cot arccot −
24
17π
75. cot (arccot (−0.001))
76. cot arccot
77. sec (arcsec (2))
4
1
78. sec (arcsec (−1))
79. sec arcsec
80. sec (arcsec (0.75))
2
√ !!
"
"√ 2 3
83. csc arccsc −
81. sec (arcsec (117π))
82. csc arccsc 2
3
√ !!
π 2
84. csc arccsc
85. csc (arccsc (1.0001))
86. csc arccsc
4
2
In Exercises 87 - 106, compute the exact value or state that it is undefined.
87. arcsin sin
π 6
π 88. arcsin sin −
3
89. arcsin sin
3π
4
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M110 Fa17
11π
90. arcsin sin
6
2π
93. arccos cos
3
5π
96. arccos cos
4
99. arctan (tan (π))
π 102. arccot cot
3
π 105. arccot cot
2
4π
91. arcsin sin
3
3π
94. arccos cos
2
π 97. arctan tan
3
π 100. arctan tan
2
π 103. arccot cot −
4
2π
106. arccot cot
3
π 92. arccos cos
4
π 95. arccos cos −
6
π 98. arctan tan −
4
2π
101. arctan tan
3
104. arccot (cot (π))
In Exercises 107
that the range of arcsecant is 0, π2 ∪ π, 3π
and that the range of
2
"assume
" - π118,
3π
arccosecant is 0, 2 ∪ π, 2 when computing the exact value.
π 4π
5π
107. arcsec sec
108. arcsec sec
109. arcsec sec
4
3
6
π π 5π
110. arcsec sec −
111. arcsec sec
112. arccsc csc
2
3
6
π 2π
5π
114. arccsc csc
115. arccsc csc −
113. arccsc csc
4
3
2
11π
9π
11π
117. arcsec sec
118. arccsc csc
116. arccsc csc
6
12
8
π "π In Exercises 119
-π 130,
assume
" π that the range of arcsecant is 0, 2 ∪ 2 , π and that the range of
arccosecant is − 2 , 0 ∪ 0, 2 when finding the exact value.
π 4π
5π
119. arcsec sec
120. arcsec sec
121. arcsec sec
4
3
6
π π 5π
123. arcsec sec
124. arccsc csc
122. arcsec sec −
2
3
6
π 5π
2π
125. arccsc csc
126. arccsc csc
127. arccsc csc −
4
3
2
11π
9π
11π
129. arcsec sec
130. arccsc csc
128. arccsc csc
6
12
8
M110 Fa17
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In Exercises 131 - 154, compute the exact value or state that it is undefined.
1
131. sin arccos −
2
3
132. sin arccos
5
133. sin (arctan (−2))
"√ 134. sin arccot 5
135. sin (arccsc (−3))
5
136. cos arcsin −
13
"
"√ 137. cos arctan 7
138. cos (arccot (3))
139. cos (arcsec (5))
"
√ !!
2 5
140. tan arcsin −
5
"√ 146. cot arccsc 5
12
149. sec arcsin −
13
144. cot arcsin
"
1
141. tan arccos −
2
143. tan (arccot (12))
12
13
5
142. tan arcsec
3
147. cot (arctan (0.25))
145. cot arccos
√ !!
3
2
148. sec arccos
√ !!
3
2
√
10
151. sec arccot −
10
150. sec (arctan (10))
3
153. csc arcsin
5
152. csc (arccot (9))
155. sin arcsin
5
13
π
+
4
156. cos (arcsec(3) + arctan(2))
3
157. tan arctan(3) + arccos −
5
13
159. sin 2arccsc
5
3
161. cos 2 arcsin
5
" √ 163. cos 2arccot − 5
"
4
158. sin 2 arcsin −
5
160. sin (2 arctan (2))
162. cos 2arcsec
164. sin
2
154. csc arctan −
3
In Exercises 155 - 164, compute the exact value or state that it is undefined.
arctan(2)
2
25
7
!!
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M110 Fa17
In Exercises 165 - 184, rewrite the quantity as algebraic expressions of x and state the domain on
which the equivalence is valid.
165. sin (arccos (x))
166. cos (arctan (x))
167. tan (arcsin (x))
168. sec (arctan (x))
169. csc (arccos (x))
170. sin (2 arctan (x))
171. sin (2 arccos (x))
x 174. sin arccos
5
172. cos (2 arctan (x))
x 175. cos arcsin
2
173. sin(arccos(2x))
176. cos (arctan (3x))
√ !!
x 3
3
177. sin(2 arcsin(7x))
178. sin 2 arcsin
179. cos(2 arcsin(4x))
180. sec(arctan(2x)) tan(arctan(2x))
181. sin (arcsin(x) + arccos(x))
182. cos (arcsin(x) + arctan(x))
1
184. sin
arctan(x)
2
183. tan (2 arcsin(x))
185. If sin(θ) =
x
π
π
for − < θ < , find an expression for θ + sin(2θ) in terms of x.
2
2
2
π
π
1
1
x
for − < θ < , find an expression for θ − sin(2θ) in terms of x.
7
2
2
2
2
x
π
187. If sec(θ) = for 0 < θ < , find an expression for 4 tan(θ) − 4θ in terms of x.
2
4
186. If tan(θ) =
In Exercises 188 - 207, solve the equation and then use a calculator or computer to approximate the
solutions which lie in the interval [0, 2π) to four decimal places.
188. sin(x) =
7
11
189. cos(x) = −
2
9
190. sin(x) = −0.569
191. cos(x) = 0.117
192. sin(x) = 0.008
194. tan(x) = 117
195. cot(x) = −12
90
17
7
200. cos(x) = −
16
√
198. tan(x) = − 10
359
360
3
196. sec(x) =
2
3
199. sin(x) =
8
201. tan(x) = 0.03
202. sin(x) = 0.3502
197. csc(x) = −
193. cos(x) =
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M110 Fa17
203. sin(x) = −0.721
206. cot(x) =
1
117
204. cos(x) = 0.9824
205. cos(x) = −0.5637
207. tan(x) = −0.6109
208. A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it touches level
ground 360 feet from the base of the tower. What angle does the wire make with the ground?
Express your answer using degree measure rounded to one decimal place.
209. At Cliffs of Insanity Point, The Great Sasquatch Canyon is 7117 feet deep. From that point,
a fire is seen at a location known to be 10 miles away from the base of the sheer canyon
wall. What angle of depression is made by the line of sight from the canyon edge to the fire?
Express your answer using degree measure rounded to one decimal place.
210. Shelving is being built at the Utility Muffin Research Library which is to be 14 inches deep.
An 18-inch rod will be attached to the wall and the underside of the shelf at its edge away
from the wall, forming a right triangle under the shelf to support it. What angle, to the
nearest degree, will the rod make with the wall?
211. A parasailor is being pulled by a boat on Lake Ippizuti. The cable is 300 feet long and the
parasailor is 100 feet above the surface of the water. What is the angle of elevation from the
boat to the parasailor? Express your answer using degree measure rounded to one decimal
place.
212. A tag-and-release program to study the Sasquatch population of the eponymous Sasquatch
National Park is begun. From a 200 foot tall tower, a ranger spots a Sasquatch lumbering
through the wilderness directly towards the tower. Let θ denote the angle of depression from
the top of the tower to a point on the ground. If the range of the rifle with a tranquilizer dart is
300 feet, compute the smallest value of θ for which the corresponding point on the ground is in
range of the rifle. Round your answer to the nearest hundreth of a degree.